If $a + 4b = 33$ and $6a + 3b = 51$, what is the value of $a + b$?
Solution: Since the problem asks for $a+b$, we look for a way to isolate $a+b$ from the given equations.


Notice that $a + 6a = 7a$ and $4b + 3b = 7b$. This gives us the key to isolating $a + b$. We simply add the two equations together:  \begin{align*}
7a + 7b &= 84 \\
7(a + b) &= 84 \\
a + b &= \frac{84}{7} \\
a + b &= \boxed{12}
\end{align*}